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Theorem cfinufil 21952
Description: An ultrafilter is free iff it contains the Fréchet filter cfinfil 21917 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
cfinufil (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem cfinufil
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elpwi 4307 . . . . 5 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2 ufilb 21930 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
32adantr 466 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
4 ufilfil 21928 . . . . . . . . . . . 12 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
54adantr 466 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐹 ∈ (Fil‘𝑋))
6 filfinnfr 21901 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑋𝑥) ∈ 𝐹 ∧ (𝑋𝑥) ∈ Fin) → 𝐹 ≠ ∅)
763exp 1112 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ 𝐹 → ((𝑋𝑥) ∈ Fin → 𝐹 ≠ ∅)))
87com23 86 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ Fin → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅)))
95, 8syl 17 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ Fin → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅)))
109imp 393 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅))
113, 10sylbid 230 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → (¬ 𝑥𝐹 𝐹 ≠ ∅))
1211necon4bd 2963 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → ( 𝐹 = ∅ → 𝑥𝐹))
1312ex 397 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ Fin → ( 𝐹 = ∅ → 𝑥𝐹)))
1413com23 86 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ( 𝐹 = ∅ → ((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
151, 14sylan2 580 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → ( 𝐹 = ∅ → ((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
1615ralrimdva 3118 . . 3 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ → ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
174adantr 466 . . . . . . . . . . . 12 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → 𝐹 ∈ (Fil‘𝑋))
18 uffixsn 21949 . . . . . . . . . . . 12 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → {𝑦} ∈ 𝐹)
19 filelss 21876 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑦} ∈ 𝐹) → {𝑦} ⊆ 𝑋)
2017, 18, 19syl2anc 573 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → {𝑦} ⊆ 𝑋)
21 dfss4 4007 . . . . . . . . . . 11 ({𝑦} ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
2220, 21sylib 208 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
23 snfi 8194 . . . . . . . . . 10 {𝑦} ∈ Fin
2422, 23syl6eqel 2858 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin)
25 difss 3888 . . . . . . . . . . 11 (𝑋 ∖ {𝑦}) ⊆ 𝑋
26 filtop 21879 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
27 elpw2g 4958 . . . . . . . . . . . 12 (𝑋𝐹 → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
2817, 26, 273syl 18 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
2925, 28mpbiri 248 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋)
30 difeq2 3873 . . . . . . . . . . . . 13 (𝑥 = (𝑋 ∖ {𝑦}) → (𝑋𝑥) = (𝑋 ∖ (𝑋 ∖ {𝑦})))
3130eleq1d 2835 . . . . . . . . . . . 12 (𝑥 = (𝑋 ∖ {𝑦}) → ((𝑋𝑥) ∈ Fin ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin))
32 eleq1 2838 . . . . . . . . . . . 12 (𝑥 = (𝑋 ∖ {𝑦}) → (𝑥𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3331, 32imbi12d 333 . . . . . . . . . . 11 (𝑥 = (𝑋 ∖ {𝑦}) → (((𝑋𝑥) ∈ Fin → 𝑥𝐹) ↔ ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3433rspcv 3456 . . . . . . . . . 10 ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3529, 34syl 17 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3624, 35mpid 44 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝐹))
37 ufilb 21930 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑦} ⊆ 𝑋) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3820, 37syldan 579 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3918pm2.24d 148 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (¬ {𝑦} ∈ 𝐹 → ¬ 𝑦 𝐹))
4038, 39sylbird 250 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝐹 → ¬ 𝑦 𝐹))
4136, 40syld 47 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ¬ 𝑦 𝐹))
4241impancom 439 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → (𝑦 𝐹 → ¬ 𝑦 𝐹))
4342pm2.01d 181 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → ¬ 𝑦 𝐹)
4443eq0rdv 4123 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → 𝐹 = ∅)
4544ex 397 . . 3 (𝐹 ∈ (UFil‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → 𝐹 = ∅))
4616, 45impbid 202 . 2 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
47 rabss 3828 . 2 ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹 ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹))
4846, 47syl6bbr 278 1 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  {crab 3065  cdif 3720  wss 3723  c0 4063  𝒫 cpw 4297  {csn 4316   cint 4611  cfv 6031  Fincfn 8109  Filcfil 21869  UFilcufil 21923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1o 7713  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fbas 19958  df-fg 19959  df-fil 21870  df-ufil 21925
This theorem is referenced by: (None)
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